Harnessing bistability for directional propulsion ofsoft, untethered robotsTian Chena,1, Osama R. Bilalb,1, Kristina Sheaa,2, and Chiara Daraiob,2

aEngineering Design and Computing Laboratory, Department of Mechanical and Process Engineering, Eidgenossische Technische Hochschule (ETH) Zurich,8092 Zurich, Switzerland; and bDivision of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125

Edited by John A. Rogers, Northwestern University, Evanston, IL, and approved April 23, 2018 (received for review January 9, 2018)

In most macroscale robotic systems, propulsion and controls areenabled through a physical tether or complex onboard electronicsand batteries. A tether simplifies the design process but limits therange of motion of the robot, while onboard controls and powersupplies are heavy and complicate the design process. Here, wepresent a simple design principle for an untethered, soft swim-ming robot with preprogrammed, directional propulsion withouta battery or onboard electronics. Locomotion is achieved by usingactuators that harness the large displacements of bistable ele-ments triggered by surrounding temperature changes. Poweredby shape memory polymer (SMP) muscles, the bistable elementsin turn actuate the robot’s fins. Our robots are fabricated usinga commercially available 3D printer in a single print. As a proofof concept, we show the ability to program a vessel, whichcan autonomously deliver a cargo and navigate back to thedeployment point.

soft robots | programmable materials | mechanical bistability |shape memory polymer | autonomous propulsion

Soft robotics (1–5) and robotic materials (also known as pro-grammable matter) (6–8) are blurring the boundary between

materials and machines while promising a better, simpler, safer,and more adaptive interface with humans (9, 10). Propulsionand navigation are core to both soft and rigid robotic systems.Autonomous (or preprogrammed) propulsion is a central ele-ment in the road map for future autonomous systems, enabling,for example, unguided traversal of open waters [e.g., studyingmarine biology (11) or ocean dynamics (12)]. Power supply toenable propulsion remains one of the major obstacles in all formsof locomotion. One of the easiest solutions for supplying powerto soft robots is the use of a tether (2). Tethered pneumatics, forexample, enabled active agonistic and antagonistic motion (13)and an undulating serpentine (14). Dielectric elastomers wereused to create tethered soft crawlers (15) and to simulate the upand down motion of a jellyfish (16). Electromagnetics were usedto create a tethered earthworm-like robot (17) and untetheredmicroswimmers under a rotating magnetic field (18). A pres-sure deforming elastomer was used to design an artificial fishtail that can perform maneuvers (19). Untethered robots, on thecontrary, sacrifice simplicity in design for moving freedom with-out restriction. An untethered robot needs to encapsulate pro-graming, sensing, actuation, and more importantly, an onboardpower source.

The demonstration of an entirely soft (composed of materialswith elasticity moduli on the order of 104− 109 Pa) untetheredrobot, “the Octobot” (4), opened the door for a new generationof robots (5). The Octobot is powered through regulated pres-sure generating a chemical reaction. Fabrication of the Octobotrequires a combination of lithography, molding, and 3D print-ing. However, it does not exhibit locomotion. A common featureof all current demonstrations of soft robots is the presence of acomplex internal architecture as a result of a multistep fabrica-tion and assembly process. Here, we present a methodology fordesigning an untethered, soft robot, which can propel itself andcan be preprogrammed to follow selected trajectories. Further-more, the robot can be preprogrammed to reach a destination,

deliver a cargo, and then reverse its propulsion direction toreturn back to its initial deployment point. The robot can befabricated using a commercially available 3D printer in a singleprint. However, the presented prototypes are partitioned to high-light the different components and speed up the printing process.

We focus on the actuation, design, and fabrication of a robotthat exploits bistable actuation for propulsion and responds totemperature changes in the environment to control its direc-tional locomotion. We use shape memory polymers (SMPs) tocreate bistable “muscles” that respond to temperature changesin the environment. Bistable actuation is often found in biolog-ical systems, like the Venus fly trap (20) and the Mantis shrimp(21). When working near instabilities, bistability can amplify dis-placements with the application of a small incremental force(22). Engineers started to integrate instabilities in design (10)[for example, in space structures (23), energy absorption mecha-nisms (24, 25), and fly-trapping robots (26)]. By amplifying theresponse of soft SMP muscles, snap-through instabilities caninstantaneously exert high force and trigger large geometricalchanges (27). Bistability has also been used to sustain a propagat-ing solitary wave in a soft medium (22). More recently, bistabilityenabled the realization of the first purely acoustic transistor andmechanical calculator (28). A typical bistability is found in theVon Mises truss design, which allows a simple 1D system tohave two stable states (29). Combining this principle with mul-timaterial 3D printing, it is possible to realize bistable actuatorswith a tunable activation force through material and geometrychanges (30). These actuators can be used to create load-bearing,

Significance

A major challenge in soft robotics is the integration of sens-ing, actuation, control, and propulsion. Here, we propose amaterial-based approach for designing soft robots. We showan untethered, soft swimming robot, which can completepreprogrammed tasks without the need for electronics, con-trollers, or power sources on board. To achieve propulsion,we use bistable shape memory polymer muscles connected topaddles that amplify actuation forces. As a proof of principle,we show that these robots can be preprogrammed to followspecific routes or deliver a cargo and navigate back to theirdeployment point. The proposed design principle can have abroad impact in soft robotics based on programed materials.

Author contributions: O.R.B. and C.D. designed research; T.C. and O.R.B. performedresearch; T.C., O.R.B., K.S., and C.D. analyzed data; and T.C., O.R.B., K.S., and C.D. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 T.C. and O.R.B. contributed equally to this work.2 To whom correspondence may be addressed. Email: kshea@ethz.ch or daraio@caltech.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1800386115/-/DCSupplemental.

Published online May 15, 2018.

5698–5702 | PNAS | May 29, 2018 | vol. 115 | no. 22 www.pnas.org/cgi/doi/10.1073/pnas.1800386115

Dow

nloa

ded

by g

uest

on

Janu

ary

22, 2

021

ENG

INEE

RIN

G

Floater

Fin

Shell

Bistable element

Shape MemoryPolymer Muscle

Pivot ofthe blade

A

C Tg = Glass transitioning temp.

B

R.T. = Room temp.

t=0s t=1.49s t=2.29s

5cm

Potential EnergyBistable Force

Rel

ease

den

ergy

Distance

Programmed state Activation Activated State

Ene

rgy

Forc

e

D

Fig. 1. Propulsion through bistability. (A) Schematic of a 3D-printed softrobot (parts are false colored for visualization). (B) Energy potential of thebistable element with two stable states: I and III. The asymmetry in thecurve indicates the need for a larger amount of energy to move back-ward than forward. The SMP muscle shown in Insets is rotated 90° withrespect to the bistable element for visualization. (C) The SMP muscle in thedeployed (I), transitioning (II), and activated phases (III). (D) Screen capturesof the deployed robot in temperature (T ≥ Tg) at the different phases ofactivation.

multistate reconfigurable 3D-printed structures, where largeshape changes are possible due to the long stroke length of thebistable actuator design. To activate them, they can be combinedwith 3D-printed shape memory strips, which respond to differ-ent temperatures, to create time-sequenced linear actuators and4D deployable structures (31). Here, we build on these works tocreate untethered robots.

Robot Design PrincipleTo show the underlying principle of our design concept, wefirst study a robot propelled by a single actuator (Fig. 1 andSI Appendix, Fig. S1). The actuator consists of two strips ofSMP material acting like a muscle connected to a bistable ele-ment. To reduce design complexity and ease prototyping, therobot is decomposed into five parts [outer shell, floaters, fins,bistable element, and shape memory muscle (Fig. 1A)]. The shell

supports the bistable element to ensure linear actuation andprovide stability for the robot. The floaters ensure that, in thevertical direction, the SMP strips are fully submerged in water.The groove in the floaters provides the pivot point for the fins.The fins are attached to the bistable element with elastomericjoints, ensuring flexibility.

The propulsion mechanism follows the forward motion ofmany organisms with fins submerged in water (32). The fins,which perform a paddling motion, are driven primarily by desta-bilizing buckling trusses. The SMP muscle used to trigger thebistable element is a pair of 3D-printed curved beams that exhibitshape memory behavior (33). One limitation of shape mem-ory materials is their slow activation speed. To overcome this,we attach the muscle to a bistable element (Fig. 1B) with thefins mounted on it. In addition to amplifying the output force(31), the bistability transforms the slow, small motion of theSMP beams into a rapid, amplified one, propelling the robotforward. It is often recognized that, in a physical implemen-tation of a bistable element using compliant joints, the twoequilibrium states are not equally energetic (i.e., |FBi,min|<|FBi,max|). Due to the rotational stiffness of the flexible joints,the fabricated state is more stable than the activated state (30).While this is a disadvantage when constructing multistable sys-tems, it can be exploited when directional transformation isdesirable (22).

Before deploying the robot, we heat the printed SMP mus-cle past its glass transition temperature (Tg) and mechanicallydeform it to the programed shape. After deploying the robot inwater (state I) with temperature equal to or larger than Tg, themuscle relaxes, transforming back into its original/printed shape(state II). Since this is a constrained relaxation, the muscle mustovercome the activation force of the bistable element at all pointsbetween states I and II [i.e., FSMP(x )>FBi(x ) ∀ x =−1, . . ., 0].After actuation, the system transitions into state III. Recentadvances in straining the material during printing may eliminatethis programing step (34). This would allow the presented robotsto function immediately after fabrication.

The vertical equilibrium of the robot is achieved by balanc-ing buoyancy and weight forces; in-plane acceleration occurs

t = 0.60 mm t = 1.00 mm

A

B

Fig. 2. Actuator design. (A) Finite element (FE) simulations of the con-strained recovery of the bistable–muscle pair. The vertical dashed black lineseparates thicknesses unable to activate the bistable element (Left) fromfunctional thicknesses (Right). (B) Experimental and numerical correlationbetween the thickness of the SMP muscle and its recovery forces as well asthe time that it takes to heat to its original shape. Insets show the differentmuscles tested. The error bars in the force readings represent the SD. Theerror bars for the activation times represent the errors in reading the timesfrom video recordings of the experiments.

Chen et al. PNAS | May 29, 2018 | vol. 115 | no. 22 | 5699

Dow

nloa

ded

by g

uest

on

Janu

ary

22, 2

021

tt

Pair 1

Pair 2

A

B

t=0s t=3.47s t=7.43s

t=0s t=3.49s t=7.42s

t=0s t=3.30s t=7.38s

I II III

C

D

E

10cm

10cm

10cm

Fig. 3. Sequential and directional propulsion. (A) A schematic of a robotwith two bistable element–muscle pairs. A thinner muscle with faster actu-ation time is placed at the rear (i.e., t1 < t2). (B) Three distinct phasesof the actuation. Activation sequence. (I) Initial state with both musclesprogramed. (II) The thinner muscle activates, triggering the first bistable ele-ment and pushing the second muscle to touch the second bistable element.(III) The second (thicker) muscle triggers the second bistable element. (C–E)Snapshots from deploying three configurations of the multistroke swimmerthat show three different directional motions at each of three phases.

when propulsion overcomes drag. The forward motion of therobot occurs in the transition between states I and III, con-sisting of pre- and postsnapping (Fig. 1B). Before the onsetof the instability (Fig. 1D), the shape memory muscle movesthe bistable element and drives the fins backward until theyare perpendicular to the direction of motion (state II). This isa relatively slow motion with low energy output. Immediatelyafter, the bistable element snaps to its second equilibrium posi-tion. This drives the fins rapidly and increases the velocity ofthe robot (Movie S1).

Bistable Element–Muscle PairTo trigger the instability, the actuation force must overcomethe energy barrier of the bistable element (Fig. 1B). To deter-mine the range of operational actuation forces, we use the finiteelement method to simulate the constrained recovery of the actu-ation pair (i.e., the bistable element and the SMP muscle) (SIAppendix, Fig. S2). We vary the thickness of the SMP beamsbetween 0.6 and 1.6 mm. Full recovery (i.e., snapping of thebistable element) does not occur for beams with thicknesseslower than 1.2 mm (Fig. 2A). Thinner beams induce localizedstresses in the bistable element, although not enough to causeit to snap. The recovery forces of the muscle range from 0.2to 2.1 N in simulation (Fig. 2B). However, we experimentallyobserve slightly larger forces at higher thickness values due to

the fusion of the two SMP beam strips at both ends of themuscle during polymerization. To determine the activation timeneeded, we simulate the actuation process using an invariantboundary temperature, while calculating the time for the mus-cles to reach thermal equilibrium through conductive heating. Tocharacterize the activation time experimentally, six SMP muscleswere first programed and then submerged in hot water simulta-neously. The time of activation is extracted from the recordedvideo (Movie S2). At equal length, the force supplied by themuscles as well as the actuation time increase with thickness(Fig. 2B).

It is worth noting, however, that the distance traveled by therobot depends mainly on the bistable element rather than themuscle. This is shown by fabricating and testing the same robotwith different muscle thicknesses (SI Appendix, Fig. S3). Regard-less of the force output of the muscle, the robot travels thesame distance (115% of its length). Such demonstration showsthe significance of the bistability and long stroke length for thepropulsion of soft robots, regardless of the initial actuation force.Since inducing a stroke depends on overcoming the bistableenergy barrier rather than the actuator force, the bistable ele-ment can be designed with a very small energy barrier (28) andhigh asymmetry (22). This is also promising for the miniaturiza-tion of the actuators and the increase of the number of actuatorsin a given robot.

Sequential and Directional PropulsionThe use of bistability for propulsion can be expanded from asingle-stroke to a multistroke robot (Fig. 3A). Multiple bistableelement–muscle pairs can be compacted into the same robotin either a connected or separate fashion. If N pairs are com-pletely disconnected, they can induce N independent strokesthat are either simultaneous (using identical pairs) or sequenced(using different ones). If the pairs are connected in series (e.g.,the muscle of pair 2 is attached to the bistable element ofpair 1), they can create synchronized strokes based on the

A

BCargo

t=0.0s t=6.16s t=8.20s t=31.1sC

Shape memory gripper

T=TL T=TL T=TL to TH T=TH

M = M

M = M

Fig. 4. Reverse navigation. (A) Schematic of a robot with two muscles anda single bistable element. The first muscle, M1, is fabricated with a materialthat activates at TL. The second muscle, M2, activates at TH, which is higherthan TL. (B) Sequence of activation. (I) At room temperature, TR.T., bothmuscles are programed. (II) As water temperature increases to TL, the firstmuscle triggers the bistable element and propels the swimmer forward. (III)As water heats to TH, the grippers relax and release the cargo. (IV) Whenthe water temperature reaches TH, the second muscle reverses both thebistable element and reprograms the first muscle. (C) Snapshots from thedeployment of the robot showing the four different phases.

5700 | www.pnas.org/cgi/doi/10.1073/pnas.1800386115 Chen et al.

Dow

nloa

ded

by g

uest

on

Janu

ary

22, 2

021

ENG

INEE

RIN

G

propagation of a soliton-like actuation (35) (Fig. 3B). This guar-antees that a stroke n will always be executed before the n +1stroke.

Postactivation of actuation pair 1, muscle 2 is moved intoposition to activate the bistable element of pair 2 (Fig. 3B,II). As the combined force of the already activated bistableelement and the muscle from pair 1 is greater than the actua-tion force of muscle 2 (i.e., FAct,1 +FBi,max >FAct,2 >FBi,min),muscle 2 displaces more in the forward direction and acti-vates the second bistable element. This creates a forward chainaction.

In addition to the increased net forward motion, due to theadded actuation pairs, propulsion in various directions can beachieved by adjusting the placement of the fins. Having two fins(one on each side) induces a symmetric moment that moves therobot forward. When one of the fins is removed, an asymmetricmoment arises, giving the robot a push toward the direction ofthe missing fin. By strategically placing the fins, navigation canbe preprogrammed in advance, setting the robot on a predefinedpath. The sequential nature of actuation allows for predictabledirectional change at each stage.

To show the sequential forward motion of the robot, four finsare placed symmetrically on the sides of two actuation pairs.The force needed to trigger the bistable element is identical tothat of the single-stroke robot. The thickness of the rear muscleis 1.2 mm, while the front one is 1.6 mm, which gives a suffi-cient time gap between the two activations. After deployed, twoconsecutive forward motions are executed, increasing the over-all distance traveled to 190% of a single-stroke robot length(Fig. 3C and Movie S3). Next, we remove the front left fin anddeploy the robot again (Fig. 3D). The rear (symmetric) actu-ation induces a forward motion followed by a left turn with≈23.85° (Movie S4). Afterward, we deploy the robot with onlytwo fins placed asymmetrically in the front left and rear rightpositions (Fig. 3D). The robot takes a left turn of ≈21.64°followed by a right one of ≈−21.45° (Movie S5). By design-ing different sizes of fins, one can control the turn angle ofthe robot.

Cargo Deployment and Reverse NavigationIn many scenarios, such as cargo deployment or retrieval, a robotis required to navigate back to the starting point of its jour-ney after the required operation is completed. The sequentialand directional motions of the robot presented so far are basedon identical muscles (i.e., same material) with varying thickness.This thickness variation translates into different activation timesat the same temperature.

To design our robot to reverse its navigation path, we incor-porate an extra muscle on the other side of the bistable element(Fig. 4 A and B). For the second set of muscles (i.e., for reversenavigation), we add an extra dimension to the design (that is,the activation temperatures). It has been shown that, by vary-ing the constituting inkjet materials, one can achieve differentglass transition temperatures (36). Using this property, we fab-ricate two identical muscles with two different materials andtherefore, activation temperatures TL and TH (Movie S6). Thesecond muscle induces enough force to overcome both the bis-table potential and the first muscle. Such force can trigger the

instability, producing a force in the opposite direction to theinitial actuation and causing the robot to reverse its navigationdirection.

To show cargo delivery and robot return, we add a shapememory gripper at the front of the robot to hold a 3D-printedpenny (Fig. 4 A and B). After the robot is deployed in waterwith temperature TL, the first muscle activates and triggers thebistable element, thus propelling the swimmer forward to thedelivery point (Fig. 4C). After the water temperature reachesthe glass transition temperature of the gripper, it relaxes to itsoriginal shape and releases the penny. When the temperaturereaches TH, the second muscle triggers the bistable element inthe reverse direction, causing the robot to move backward to itsoriginal deployment point (Movie S7). The principle of reversepropulsion can be easily extended to more complex trajectoriesand multiple cargoes. Future improvements of this technologycould include the use of prestrained materials (34), reversibleSMPs (37), or liquid crystal elastomers (LCEs) (38, 39) in themuscles. LCEs activated by light (38) or temperature variations(39), for example, could achieve cyclic actuation in response todiurnal/nocturnal cycles.

OutlookThis work presents a 3D-printable swimming robot that requiresneither complex onboard components nor a tether to achievedirectional propulsion. Instead, programed SMP muscles areused as the power supply, a bistable element is used as the“engine,” and a set of fins is used as the propellers. By reactingto varying external temperatures, the robot can move forward,turn, reverse, and/or deliver a cargo. At constant temperature,the robot response time can be controlled by varying the geome-try and material properties of both the bistable elements and theshape memory strips. This shows a first step in the realization ofsoft locomotive robots that are potentially applicable in a varietyof applications, such as navigation and delivery.

Materials and MethodsAll components are fabricated with a multimaterial Stratasys Connexprinter. The SMP muscles are fabricated with VeroWhitePlus plastic (Tg'60◦), with the exception of M1 in Fig. 4, which is fabricated withFLX9895 (Tg' 35◦). The compliant components within the bistable ele-ment are fabricated with Agilus30 with Tg <−5◦ and therefore, remainin the rubbery state throughout the experiments. All remaining compo-nents are fabricated with a high temperature-resistant material, RGD525(Tg > 80◦), which retains its stiffness throughout the activation process(33). The stiffness values of the materials used in the actuator are1 GPa (VeroWhitePlus) and 1 MPa (Agilus30), which fall within therange of soft materials (104− 109 Pa) as defined in ref. 4. The dimen-sions for the water container used to show the single and sequentialmotion are 967 mm (length)× 555 mm (width)× 393 mm (height). Thewater is heated by an Atwood heating element, which keeps the watertemperature consistently above the glass transition temperature of theused materials. Finite element simulations are implemented using Abaqus14.1 (geometrical nonlinear solver with thermal–viscoelastic materialmodel).

ACKNOWLEDGMENTS. We thank Jung-Chew Tse for fabrication support andConnor McMahan and Ethan Pickering for support in experiments. Thiswork was supported by Army Research Office Grant W911NF-17-1-0147 andEidgenossische Technische Hochschule (ETH) Postdoctoral Fellowship FEL-2615-2 (to O.R.B.).

1. Kim S, Laschi C, Trimmer B (2013) Soft robotics: A bioinspired evolution in robotics.Trends Biotechnol 31:287–294.

2. Rus D, Tolley MT (2015) Design, fabrication and control of soft robots. Nature521:467–475.

3. Wang L, Iida F (2015) Deformation in soft-matter robotics: A catego-rization and quantitative characterization. IEEE Rob Autom Mag 22:125–139.

4. Wehner M, et al. (2016) An integrated design and fabrication strategy for entirelysoft, autonomous robots. Nature 536:451–455.

5. Mazzolai B, Mattoli V (2016) Robotics: Generation soft. Nature 536:400–401.

6. McEvoy MA, Correll N (2015) Materials that couple sensing, actuation, computation,and communication. Science 347:1261689–1261689.

7. Goldstein SC, Campbell JD, Mowry TC (2005) Invisible computing: Programmablematter. Computer 38:99–101.

8. Bilal OR, Foehr A, Daraio C (2017) Reprogrammable phononic metasurfaces. AdvMater, 29:1700628.

9. Zheludev NI, Kivshar YS (2012) From metamaterials to metadevices. Nat Mater11:917–924.

10. Reis PM, Jaeger HM, van Hecke M (2015) Designer matter: A perspective. ExtremeMech Lett 5:25–29.

Chen et al. PNAS | May 29, 2018 | vol. 115 | no. 22 | 5701

Dow

nloa

ded

by g

uest

on

Janu

ary

22, 2

021

11. Fernandes PG, et al. (2000) Fish do not avoid survey vessels. Nature 404:35–36.12. Jaffe JS, et al. (2017) A swarm of autonomous miniature underwater robot drifters

for exploring submesoscale ocean dynamics. Nat Commun 8:14189.13. Shepherd RF, et al. (2011) Multigait soft robot. Proc Natl Acad Sci USA 108:20400–

20403.14. Onal CD, Rus D (2013) Autonomous undulatory serpentine locomotion utilizing body

dynamics of a fluidic soft robot. Bioinspir Biomim 8:026003.15. Henke EFM, Schlatter S, Anderson IA (2016) A soft electronics-free robot.

arXiv:1603.05599v1.16. Godaba H, Li J, Wang Y, Zhu J (2016) A soft jellyfish robot driven by a dielectric

elastomer actuator. IEEE Rob Autom Lett 1:624–631.17. Song CW, Lee DJ, Lee SY (2016) Bioinspired segment robot with earthworm-like plane

locomotion. J Bionic Eng 13:292–302.18. Tottori S, et al. (2012) Magnetic helical micromachines: Fabrication, controlled

swimming, and cargo transport. Adv Mater 24:811–816.19. Marchese AD, Onal CD, Rus D (2014) Autonomous soft robotic fish capable of escape

maneuvers using fluidic elastomer actuators. Soft Robot 1:75–87.20. Skotheim JM, Mahadevan L (2005) Physical limits and design principles for plant and

fungal movements. Science 308:1308–1310.21. Patek S, Korff W, Caldwell R (2004) Biomechanics: Deadly strike mechanism of a

mantis shrimp. Nature 428:819–820.22. Raney JR, et al. (2016) Stable propagation of mechanical signals in soft media using

stored elastic energy. Proc Natl Acad Sci USA 113:9722–9727.23. Schioler T, Pellegrino S (2007) Space frames with multiple stable configurations. AIAA

J 45:1740–1747.24. Restrepo D, Mankame ND, Zavattieri PD (2015) Phase transforming cellular materials.

Extreme Mech Lett 4:52–60.25. Shan S, et al. (2015) Multistable architected materials for trapping elastic strain

energy. Adv Mater 27:4296–4301.26. Zhang Z, Chen D, Wu H, Bao Y, Chai G (2016) Non-contact magnetic driving bioin-

spired venus flytrap robot based on bistable anti-symmetric CFRP structure. ComposStruct 135:17–22.

27. Overvelde JT, Kloek T, D’haen JJA, Bertoldi K (2015) Amplifying the response ofsoft actuators by harnessing snap-through instabilities. Proc Natl Acad Sci USA 112:10863–10868.

28. Bilal OR, Foehr A, Daraio C (2017) Bistable metamaterial for switching and cascadingelastic vibrations. Proc Natl Acad Sci USA 114:4603–4606.

29. Mises RV (1923) Uber die Stabilitatsprobleme der Elastizitatstheorie. Z Angew MathMech 3:406–422.

30. Chen T, Mueller J, Shea K (2017) Integrated design and simulation of tunable, multi-state structures fabricated monolithically with multi-material 3D printing. Sci Rep7:45671.

31. Chen T, Shea K (2017) Autonomous deployment of space structures througha shape memory polymers triggered bistable actuator. 3D Print Addit Manuf ,in press.

32. Song SH, et al. (2016) Turtle mimetic soft robot with two swimming gaits. BioinspirBiomim 11:036010.

33. Wagner M, Chen T, Shea K (2017) Large shape transforming 4D auxetic structures. 3DPrint Addit Manuf 4:133–142.

34. Ding Z, et al. (2017) Direct 4D printing via active composite materials. Sci Adv3:e1602890.

35. Nadkarni N, Arrieta AF, Chong C, Kochmann DM, Daraio C (2016) Uni-directional transition waves in bistable lattices. Phys Rev Lett 116:244501.

36. Mao Y, et al. (2015) Sequential self-folding structures by 3D printed digital shapememory polymers. Sci Rep 5:13616.

37. Behl M, Kratz K, Zotzmann J, Nochel U, Lendlein A (2013) Reversible bidirectionalshape-memory polymers. Adv Mater 25:4466–4469.

38. Palagi S, et al. (2016) Structured light enables biomimetic swimming andversatile locomotion of photoresponsive soft microrobots. Nat Mater 15:647–653.

39. Kotikian A, Truby RL, Boley JW, White TJ, Lewis JA (2018) 3D printing of liquidcrystal elastomeric actuators with spatially programed nematic order. Adv Mater,30:1706164.

5702 | www.pnas.org/cgi/doi/10.1073/pnas.1800386115 Chen et al.

Dow

nloa

ded

by g

uest

on

Janu

ary

22, 2

021